Bounded Populations. Extinction Time and Time of Supplanting All Particles by Particles of One Type.

Klokov S.A., Topchii V.A.

Omsk Branch of Sobolev Institute of Mathematics SB RAS

Object• Population dynamics and population evolution analysis based on changes of the DNA. • Discrete time Markov models. • Haploid models without mutations.• Population size is fixed (the birth and death distribution is such that the total population size does not change) or bounded. • Random number of offspring. • Evolution of families. • Extinction time.

The most popular models of the given class were introduced by Wright and Fisher, Moran, Karlin, and McGregor. In the Wright–Fisher model, the numbers of particles of different types in a generation are the parameters of a polynomial distribution of the birth of offspring. In the Moran model, only one particle dies replacing one or several others with its offspring. In the Karlin and McGregor model, the process branches and the distribution is redefined by the condition of fixed total number of offspring.

Wright–Fisher Model

k

N-kN-j

j

. . . . . . . . . . . .

All parents are equally likely

Z Z Z Z Z Z Z Z0 1 2 3 4 n- n-1 nn

n

EZn=m^nZubkov 1975 Galton-Watson

m=E<1)(=1)(>1) ; Z >0n

Galton-Watson. Common ancestor.

(n)m<1; (n)nm>1; (n) nuniformly on [0,1] m=1.

Common Ancestor• Branching modelsO’Connell Neil (1995) 750000-1500000 years ago• Wright–Fisher model (Mitochondrial Eve)unisexual Cann (1987) 100000-200000 years agoIn Africabisexual Сhang (1999) 500 years ago , 32

generations

Fixation. Example 1.

Population size 5; Maximal number of offspring 3; 4 generations passed.

Fixation. Example 1. s=2.

•

O

O

O

O

O

OO

O

O O

OO

OO O

OOOOO 3nZ1nZ2nZ3nZ

Empirical Results

Theorem 1.

Let be fixation time, N population size, k number of type I individuals initially, s>1 splitting factor, then

)ln()(lnln1

21

kNkNkkNNsNsE Nk

Empirical Results

Suppose that we have a fixed size population consisting of N particles each of which can belong to one of N types. The composition of the population changes at the integer time moments. If, at the present time moment, the number of particles of each type is defined by the vector

then, at the next time moment, the number of particles of each type is described by the random vector

We assume that the last random vectors have exchangeable distributions (i.e., the birth and death law does not depend on their type) are independent and identically distributed

We are interested in the random variable , the fixation time of the population (i.e., supplantingall particles by particles of one type or the first attainment of the set of absorbing states )and a bound for its expectation in terms of the moments of the distribution Along with , we consider the random variable equal to the number of generations startingfrom which the population first consists of particles of a single type.

Theorem 2.

0 time

Population size

Upper bound for population size

…

Initi

al

Popu

latio

n si

zeExtinction

Population size trajectory. Time to extinction is Huge

Extinction

Let be supercritical branching process, ,

- probabilistic generating function.

Let and - extinction probability.

- - population bounded on level l. Let

Theorem 3.

Let and ,then

For crush period